Problem 90
Question
Find the standard form of the equation of the parabola with the given characteristics. Focus: (0,0)\(;\) directrix: \(y=8\)
Step-by-Step Solution
Verified Answer
The standard form of the equation of the given parabola is \(y=-4x^2\).
1Step 1: Identify Focus and Directrix
The given focus is at (0,0), and the equation of the directrix is \(y=8\).
2Step 2: Define the Vertex
The vertex of the parabola is halfway between the focus and the directrix. This makes the vertex also at (0,0), the same as the focus.
3Step 3: Find the value of 'p'
The value of 'p' in the standard equation of a parabola represents half the distance between the focus and directrix. Here, \(p=-4\), negative because the parabola opens downwards.
4Step 4: Write the Standard Form of the Parabola
Our parabola is vertical and has a vertex at the origin (0,0), so its equation according to the general formula \(y=a(x-h)^2 + k\) will be simplified to \(y=px^2\), substituting \(p=-4\), the equation of the parabola becomes \(y=-4x^2\).
Other exercises in this chapter
Problem 89
Find the standard form of the equation of the parabola with the given characteristics. Focus: (2,2)\(;\) directrix: \(x=-2\)
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